I'm posting one puzzle, riddle, math, or statistical problem a day. Try to answer each one and post your answers in the comments section. I'll post the answer the next day. Even if you have the same answer as someone else, feel free to put up your answer, too!
Friday, September 28, 2007
Another Counting Problem
In how many different ways can 6 objects be arranged in a circular pattern?
Well, let's look at how many ways they can be arranged in a linear pattern. This is a classic factorial problem, so the answer to that is 6!
However, when you make the pattern circular, for each pattern that you have, the "starting number" is arbitrary. This means that 123456 = 234561 = 345612 = ...
So, each pattern with "1" as the starting number takes care of 5 others (1 pattern -> 6 patterns). So, I'd just divide by 6.
Circular permutations. From a placement point of view, ask the first person to sit. Doesn't matter where. Now, the second person has to make a choice, and the third, etc. 1 x 5 x 4 x 3 x 2 x 1
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Well, let's look at how many ways they can be arranged in a linear pattern. This is a classic factorial problem, so the answer to that is 6!
ReplyDeleteHowever, when you make the pattern circular, for each pattern that you have, the "starting number" is arbitrary. This means that 123456 = 234561 = 345612 = ...
So, each pattern with "1" as the starting number takes care of 5 others (1 pattern -> 6 patterns). So, I'd just divide by 6.
6!/6 = 5*4*3*2*1 = 120
I think you've figure it out Abe. It actually wasn't the answer I was thinking of, but your answer makes a lot of sense.
ReplyDeleteCircular permutations. From a placement point of view, ask the first person to sit. Doesn't matter where. Now, the second person has to make a choice, and the third, etc.
ReplyDelete1 x 5 x 4 x 3 x 2 x 1
Ofcourse the answer is gonna be 120 since we would have to find the answer by keeping one object fixed and then finding the possible ways.
ReplyDelete